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Boxes, extended boxes and sets of positive upper density in the Euclidean space

Part of: Harmonic analysis in several variables Classical measure theory Sequences and sets Extremal combinatorics

Published online by Cambridge University Press:  14 January 2021

POLONA DURCIK  and
VJEKOSLAV KOVAČ
POLONA DURCIK
Affiliation:
California Institute of Technology, 1200 E California Blvd, Pasadena CA 91125, U.S.A. Current address: Chapman University, One University Drive, Orange, CA92866, U.S.A. e-mails: durcik@chapman.edu, durcik@caltech.edu
VJEKOSLAV KOVAČ
Affiliation:
Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia e-mail: vjekovac@math.hr
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Abstract

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We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: 2n vertices of a fixed n-dimensional rectangular box, the same vertices extended with n points completing three-term arithmetic progressions, and the same vertices extended with n points completing three-point corners. Our results provide common generalizations of several Euclidean density theorems from the literature.

MSC classification

Primary: 05D10: Ramsey theory 28A75: Length, area, volume, other geometric measure theory
Secondary: 11B30: Arithmetic combinatorics; higher degree uniformity 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 3 , November 2021 , pp. 481 - 501
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Cambridge Philosophical Society 2021

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